Transcendental number: Difference between revisions
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In [[mathematics]], a '''transcendental number''' is any [[complex number]] that is not [[algebraic number|algebraic]], i.e. it is not a root of any [[polynomial]] whose coefficients are [[integer]]s, or, equivalently, it is not a root of any polynomial whose coefficients are [[rational number|rational]]. | In [[mathematics]], a '''transcendental number''' is any [[complex number]] that is not [[algebraic number|algebraic]], i.e. it is not a root of any [[polynomial]] whose coefficients are [[integer]]s, or, equivalently, it is not a root of any polynomial whose coefficients are [[rational number|rational]]. | ||
Transcendental numbers are necessarily [[irrational number|irrational]], but there are many irrational numbers that are not transcendental. For instance, <math> \sqrt{2} </math> is irrational. However it is algebraic, since it is a root of the polynomial <math> x^2-2 </math>. It is thus irrational but not transcendental. | |||
Proving a number to be transcendental is generally much more difficult than just proving it is irrational. Examples of real numbers known to be transcendental are [[pi|<math>\pi</math>]] and [[e (mathematics)|<math>e</math>]]. | |||
Revision as of 11:27, 8 May 2008
In mathematics, a transcendental number is any complex number that is not algebraic, i.e. it is not a root of any polynomial whose coefficients are integers, or, equivalently, it is not a root of any polynomial whose coefficients are rational.
Transcendental numbers are necessarily irrational, but there are many irrational numbers that are not transcendental. For instance, is irrational. However it is algebraic, since it is a root of the polynomial . It is thus irrational but not transcendental.
Proving a number to be transcendental is generally much more difficult than just proving it is irrational. Examples of real numbers known to be transcendental are and .
